1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Interpolation and linear algebra

  1. Dec 3, 2014 #1
    1. The problem statement, all variables and given/known data
    As a string in my program.

    2. Relevant equations
    Solving a system with the forward phase of row echelon reduction and a consecutive back substitution.
    All done by numpy here. (The book suggested MATLAB, etc).

    3. The attempt at a solution
    Code (Python):
    import numpy

    """ In a wind tunnel experiment, the force on a projectile due to air
    resistance was measured at different velocities:
    Velocity (100 ft/sec) 0 2.00 4.00 6.00 8.00 10.00
    Force (100 lb)        0 2.90 14.8 39.6 74.3 119.0
    Find an interpolating polynomial for these data and estimate the force on the
    projectile when the projectile is traveling at 750 ft/sec.
    Use p(t) = a0 + a1*t + a2*t^2 + a3*t^3 + a4*t^4 + a5*t^5.
    What happens if you try to use a polynomial of degree less than 5?
    (Try a cubic polynomial, for instance.)
    """


    make_row_for_t = lambda t: [t ** n for n in range(6)]

    A = numpy.array([make_row_for_t(t * 100) for t in range(0, 11, 2)])
    B = numpy.array([[n] for n in [0, 290, 1480, 3960, 7430, 11900]])

    coefficients = numpy.linalg.solve(A, B)

    p = numpy.polynomial.Polynomial(coefficients.flatten())

    print_p = lambda x: print("{:<8} = {}".format("p(" + str(x) + ")", p(x)))

    for n in range(0, 1100, 200):
        print_p(n)

    print_p(750)
    Code (Text):
    p(0)     = 0.0
    p(200)   = 290.0000000000015
    p(400)   = 1480.0000000000023
    p(600)   = 3960.0000000000036
    p(800)   = 7430.000000000004
    p(1000)  = 11900.00000000001
    p(750)   = 6483.837890625005

    Did it work? Yes. (Just get the python3 and python3-numpy packages under Linux and it should run in your machine too.)

    The book does not have the answer, but mine seems OK, I wanted to get some review on what I did as I am new to numpy and interpolation and thought I would get some useful advice or even corrections.
     
  2. jcsd
  3. Dec 3, 2014 #2

    NTW

    User Avatar

    The interpolation requires a polynomial in x5, as you have six data points. For 750 fps, you get 6483,83 lbs

    With a smaller polynomial, and if you want to consider all the points, you would have to conform yourself with a best-fit (that, in this case, is quite good with a third-degree polynomial...).Since you are asked for the resistance at 750 fps, the sensible thing to do, in my opinion, is to take the last four points and to obtain the interpolation, third-degree polynomial.
     
  4. Dec 3, 2014 #3

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    Very few things are interpolated well using polynomials higher than third order. You fight round-off errors and then the polynomial will want to oscillate. Most data interpolate quite well with either parabolas or cubics.
     
  5. Dec 3, 2014 #4
    Interesting. Thank you for the answers, will have a somewhat decent graph of the cubic soon. Maybe will compare it with the fifth degree polynomial. My first thought was that really small coefficients and bigger polynomials would just make the interpolation more and more accurate.
     
  6. Dec 3, 2014 #5

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    If the coefficients of the higher powers of your interpolating polynomial are really small, that can also be an indication that a lower order polynomial will interpolate the data just fine.
     
  7. Dec 3, 2014 #6
    p(t) = 2.0833e-07*t^3 + 0.012*t^2 + 0.24167*t^1 - 550 // using the last 4 points.
    p(t) = 0.0125*t^2 - 0.15*t^1 - 450 // using the last 3 points.
    Nice that the graphics (in the region of interest) almost coincide.
    upload_2014-12-4_0-12-20.png
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Interpolation and linear algebra
  1. Interpolation error. (Replies: 1)

Loading...